iranian 9th grade coursebook : math

7/23/2019 Iranian 9th grade coursebook : Math

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ISBN 978 -964 - 05 - 2435-0 9

-964-05 - 2435-0


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1 ........................................................

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15 ..................................

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:........................................ 44

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59 .......................................... ..........

60.........................................................:

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68........................................................:

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............................................113

:.....................114

119.................................. :

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130..............................................

131...........................................

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135........................................... :

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9/151

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10/151

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{A = {2,3,5 B A

A

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A

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B

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C

D

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60

3

{4,6,10} :{6,10,12} :

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11/151

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12

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A = {

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,5,7}

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A = {a,b,5,7} A

A a aA A a

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A

4A

A

:

A = {,2,,4}

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1 2

4 3

A

B

A

1

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:B = {,,,} {,,,4,,2} = A 2

A

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B

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A B

ab

c

fs

k

m

n


12/151

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24 ( ( )

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b

a

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E ={ } )5

F= {2,4,6,8} )6

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{3,a,b})7

H={2,3,5,7} {6,4,2,8})8

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13/151

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C = {10} )

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{2,3,4,...,9}

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A = {3,5,7,9,11}

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.________ A

.2 3


14/151

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A

:

10 12

4 2

B

B

A

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A

B

B

A

B A

A B B A

. A = B

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A

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A B B A

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2 9 2 1445 4 3 25

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15/151

D

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B

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A

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16/151

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B A , 3 B , A B , B C , A C , 2 A

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B

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17/151

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{ }, , , , , . . .= 1 2 3 4 5

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E = {2,4,6, 8 ,...}

2k

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{ }A x x= <


18/151

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{ }A x | x , x=


19/151

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{ }A x | x= 6

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.)

AB

B

A

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A

AB . B

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{ }A B x | x A, x B=

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AB .B A

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:

A B

A B

A B

A B


20/151

B A :

:

AB

AB

{ }A , , , , ,= 1 2 3 4 5 8 { }B , , , ,= 3 4 5 6 7

{ }A B , ,= 3 4 5 , { }A B , , , , , , ,= 1 2 3 4 5 8 6 7

A B

1 2

8

6

7

3

4

5

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B

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21/151

( AC ( BC ( C)A B(

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B

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1

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:

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B - A

A - B

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A B B A

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{ }B c,d,k, f , s, t=

{ }A a,b,c,d,e,k =

:

{ }A B a,b,e = { }B A f ,s, t =


22/151

A B

1

2 3

4

59

6

7

{ }C , , , ,= 1 7 8 1011

{ }B , , , ,= 1 5 7 3 9

{ }A , , , ,= 2 4 6 8 9

:

( AB ( BC ( A C ( AB

( A - B

( C - B

()A - C( )B - C(

()AB( - C

( AA

( A

( BB

( C

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{ }(A B) (B A) , , = 12 6 (

n )A B( = 8 )

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23/151

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{ } { }A , , S , , , , , ; n(A) , n(S)= = = =3 6 1 2 3 4 5 6 2 6

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2 1

6 3

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n(B)P(B)

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= = =3 1

6 2

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; C n( ) 0= = (

P(C) P( )= = =0

06

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{ }; D , , , , , S= =1 2 3 4 5 6 (

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61

6

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24/151

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2

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/2 1 4 /3 1 7 /5 2 2 /6 2 4 / /7 2 6 8 2 8


38/151

:

| | ( ) = = + = 1 2 1 2 1 2 2 1

:

1 2

/2 1 4 :

1( | | =2 3 :

2( | | =7 8 :

3( | |2 5 5 :

4( | | =4 3 :

:

|a + b + c|

c = -3

b= 2

a =1

2

:

| a b c | | ( ) | | / |+ + = + + = +1

2 3 2 5 22

( / )2 1 4 2 5 2+ /

. / 2 5 2 -) / +2 5 2(

| | | | ( ) ( ) + = 3 5 2 5 3 5 2 5 :

= + + =3 5 2 5 5

:

a ( ) ( ) ( ) ( )

2 2 2 2 2 2 2 23 3 6 6 7 127 325

3

a | a |=2 :

: ( ) 21 3 :

( ) | | ( ) = = = +21 3 1 3 1 3 1 3


39/151

:

( |)-7(2| |-7|2

|( |-8 + 5| |-8| + |5

|( |3-9| |3| - |9

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40/151

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41/151

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.ABCD :

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47/151

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AD

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ADC ABC

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D C

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1

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48/151

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PB

PA

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180

60

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180

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ABC

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A B C A A A

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BC

1 2

3


49/151

:

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d

a

b

12

3 4

56

78

A

B C

1

2

3

d ABC

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A

AB A A23 2 1

C A

= 3 AC B A

= 1

A B C A A A

+ + = + + =2 1 3 180 : C

B

A

3 A

1

:

d| |BCB A

ABA B C A A A

d| |BCC A

AC

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=

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2 1 3

3

180

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.5000 5000

.


50/151

. :

: O

3 O

1

O OO O O O O O

O O

+ =

+ = + =+ =

1 2

1 2 3 2 1 3

3 2

180

180

1

2 3

O

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:

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ABC

:

180

60 C

B

A

1

A A A A

+ = = = =1 2 2 1180 180 180 60 120

B C

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= +2

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A

B

1

2


51/151

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52/151

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AB A B

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53/151

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:

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:)( OC=OD OA=OB :

)( O

2 O

1 .AD=BC

OBC OAD

:

OA OB

OC OD OBC OAD AD BC

O O

=

= =

= 1 2

O

A C

D B

2 1

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)A

(

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10 )B

(

C

D )BC=6( 6

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.AC=BD

.

m10A B

CD

D C

BA


54/151

BD AC . D

C

1

AC=BD :

.AC=BD

ADB

ACB

N ABCDM

ADM ABN

.CB CD

:

1

AD = AB =____=____ , BN =____

A

=____, B

=____ , DM =____

: ADM ABN

ADM

ABN

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2.

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A

B

C

M N

D

.

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AB

CD


55/151

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CD

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BD

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=1 1

AB =____ : B D

=2 2

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AC

_____

___

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_____

____ ____


56/151

.

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1

.

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OA = OC :

( .

2

!(

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AM

AMC

AMB

BC AM A

MBMA M 4

.

( . :

).B A M

d

5

d B A AB

.OH=OH

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(

:OH=OH

OA OB

O O OAH OBH OH OHAH BH

=

= ==

1 2

)(

)(

)(

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A

O

B

CD

A

B CM

M

A

O

B

A

H

O

d

B1

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H


57/151

.

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A

:

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B

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M

d

B

H

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58/151

)( )( 3

BH

AH MA=MB AB M .

AH=BH

:d

MA=MB , H H

= = 90

AH=BH

. 4

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)( ( (MA MB

H H AH BH

( )

=

= = ==

90

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CD

AB

. CD AB 1

. CD AB

.CD AB

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3

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O

B

A

D

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O

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A

D

C

____ __ __

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A

D

C


59/151

HO

B

A

D

C

H

AO

B

C

D

OA OD( )

H H OAH AH

( )OH OH AH AB CD

= =

= = = = = =

2 2

AB=CD

AB CD= 1

: 2

AB CD

BC BC

AB BC CD BC

=

=

=+ = +

AC=BD 3

B

A D

C

ABCD

Q

P

N

M

MN=PQ :

AD BC O 2

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BC

AD

A M

NQ

B

CPD

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CD AB 4

D A .)OH=OH

(

CD

AB

:


60/151

N M ABC 3

.BM=NC BC

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AMN

AF

BE

ABCD

4

B1

A1

AF. BE

5

:

.MB=MC

A

B CM N

11

D F E C

A B

A

B CD

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61/151

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64/151

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86

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4 3

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1 2

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65/151

: 2

A = 12

B =

10

C = 30 ABC

A =

3

2 B

=

3

6 C

=

7

2ABC

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A B C

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LK

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66/151

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ABC 8

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B C

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67/151

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68/151

:

= )-5(=5

=

43 81

4 256

=

51 1

2 32

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a

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)

(

:

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:

=5

1 1

32 2

=4

1 1

16 2

=3

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8 2=

2

1 1

4 2

1

2124816

222222021222324

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.)

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n

na

a

= 1

a , nN

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1 17

497

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= = =

4

4

2 1 1 81

163 162

813

(

= = =

2

21 1 1 25

15 1

255

( ( )( )

= =

3

3

1 1282


69/151


70/151

am* an= am+n: a n m

.

= = = =6

4 6 6 6 4 2

4 4

1 33 3 3 3 3

3 3

+ = = = =

5 2 7

5 2 5 2 71 1 1 12 2 2

2 2 2 2

( ) ( ) ( )

= = =

3 5 8

3 5 81 1 12 2 2

2 2 2

:

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)

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a

n

m

:

am* an= am+n

:

23*2-5*2-4=23-5-4=2-6

)2x-1(*)3x4(*)4x3(=24x-1+4+3 =24x6 )x0(

)b , x, y0( :

5-7*510= )-4(-9*)-4(-1=

=

4 93 3

8 8( ) ( ) =4 22 2

b-2*b-3=x x

y y

7 11


71/151

.

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9

3

6

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1

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8

: n m b a

mm n m n

n

aa a a

a

= = ; mm

aa

= 1

;mm

m

a a

bb

=

;

)am(n=amn ; )ab(m= am.bm ; a=

)

=3

5

7

7

=( -*5-

)

=

3

3212

3

)

=

12

2

5

(

= =

8 10 8 10

4 6 4 6

2 5 2 5

2 5 2 5

)

( )x .y .z x .y .z

x .y .z

= =5 2

5 2 2 7 1 3

2 7 3 x,y, z


72/151

.-

.

:

=

> ) = 2 2

66

( -=- ( -> -

.

)

10

3127

3 ( )/(-* 5-

(

4 415 45

14 28

( )-5-(-

.

-

5-

21

2

)-(

)-(

-

--

.

)/(> )/(> )/(

25

4 > )/(

23

4< )/5(

.

)

3 3

5 8

2 8

3 3

2 2

)

12

2

3


73/151

.

105

105

104

104

1000

1000

100

100

10

10

/

2

9/3

. 300000000 2

.

.

= 600 0000000060000 =

= 60000

.

:

/ / /

/

=

= =

=

8

8 5 13 14

5

300000000 3 10

3 10 3 6 10 10 8 10 1 08 10

360000 3 6 10

.

.)(

:


74/151

:

=/*5 =/*

=/ * =/*

5/ =/5 *

/.

: 0/000007

0/000007 = 710-6

/

.

1/610-3

a 10n

.n 1a > 10

0/0000127

= 1/27

10- 123= 1/23102 :

0/0137 = 1/37 10-2 29000= 2/9104

: 1

24000 = 10000000 =

0/00

= 0/000061 =

1404 = 0/127

=

:

2

/2 * 10-3= 7/304 * 10-=

2/28 * 108 = 9/4612 * 109=

6/02 * 10-2= 1/1 * 104=


75/151

: 1

)

5 5

5 5

3 10 25

4 15

)

1 2

4

8 4

12

8

2

1/02 * 10-= 0/00000102 /9 * 10-1=0/9

4/3 * 10

3

= 4300

7/004 * 10

-2

=0/7004

6/18 * 107= 61800000 8/2

70 * 104= 82

70

.

5

3

. /5 4

.1/3* 1071/4 * 109

:

6

2 * 10-7 * 4 * 109 /5 -4

___________

5 -19

6/287 * 1089/17 * 107 7

.

.

3

8

/ * < / / < / *

: 9

/5- , / , 5/5- , /-


76/151

:

)-(= ( ) =25 5 =

21

7

=

22

3

( ) =25

=

21

7= )-(=

.

-

.

-

)

(

:

9

9

. =9 3 = 9 3

:

552

3

2

3-33

6149

169

) (

7 7 . 2

3

2

3

4

9

. =0 0 .

b b b

.

.

b

.

02

35

1

2

1

5-33-22

64-88

)

(

:


77/151

8 .= )(

. =3 8 2 8 3

= 3 8 2

-

-

)-(=-

.

. 8

27

:

( ) =3 38 =3

1

8 =3125 =3 27

.

b3 b

.

:

=81 =24 ( ) =24 =3 1

=3 27

125 =3 36 =3

8

1000 ( ) =33 7

: x | x |=2

( ) =26 =28

=

23

5

( ) =

2

1 2 ( ) =

2

2 9

=

21

13

x y+2 2

.

x y+ =2 2 .)x( y ( x


78/151

:

b

a

ab a b= a ab b

=

:

.b a

a b3 33 ababb3ba3a

2*5=10101000512528

1

8

27

27-8

ab a b= 3 3 3 : b a

:b

a a

b b=

3

3

3

.

+ = +3 3 38 27 8 27

.

.4 5


79/151

:

49

16,

1

81, , , ,

:

, , - , 1

216

,

( ) = 21 1 ( ) = 33 1 1 ( ) | | = =25 5 5 ( ) = 33 5 5

= 49 7

256 16

/ /=1 44 1 2

( ) =21 1

= 3 64 4

:

3 125 36 3 1 81 381

3 3 325 5

-

-

:

= =3 3

4 16 = =

33 33 2 5 4 15 8

= =33 3 3

128 64 4 2 = 33 3

20 4

= =

3

3

3

125 125

64 64

= = =3

33

3

54 5427

22

___

___

___

___

______


80/151

a

:

a x <

= = =2 2

90 2 3 5 3 10 3 10 :

150 , 80 , 24 ,3 2125

( ) = 33 2 2 = 3 34 4

:

=3 32 16 3 4

=8 5

10

=

3 3

3

18 60

5


81/151

. 2 6

.

.

x x ( )+ =2 2 22 6 :

________

:

x= x= :

. ________

2 3 12 ________

.

________= + + + = =2 3 2 3 2 3 2 3 4 2 3 8 3

:

7 2 3 2

+ =7 2 3 2 10 2

=7 2 3 2 4 2 :

:

+ = + =12 9 3 2 3 9 3 11 3

.

32 7 2

32 2 5

:

x

x

:

2xya35

2

332

2

3 7

3x2b22 524 5

5x7a1

52

103

38 28 7

6 2xy4b72 10 35 22 5

+39 2 22

11 7 2 5


82/151

.

.

) +72 32 18

:

).(

____272 6 2 6= =

____ ____232 4 4= =

____

18 3 2= =

____ _______ _______

72 32 18 6 5 2 + = + = :

( + + = + + 332 3 33 3

50 24 81 5 2 2 3 3 3

____ ____ ____ ____ ____3 3 3

5 2 3 5 5= + + = +

. ( )+48 3 2 :

) ( )+ = + = + 2 2 248 3 2 48 3 48 2 4 3 4 3 2

( )= + = +24 3 4 6 12 4 6

) ( ) ( ) ( )+ = + = + = +248 3 2 4 3 3 2 4 3 3 2 12 4 6

.

( )+ 12 27 48 3 :2

+ = + =12 27 48 2 3 3 3 4 3 3

1( .(


83/151

.

( +98 50 128 ( + +1 4

481 9

( +27 12 75 48

( + 3 3 35 2 3 54 4 128 ( ( )( )+ 2 3 3 2 3

20

2

:

2

= = =20 20 2 20 2

10 222 2 2

.

.

( = =5 5 5 3

62 3 2 3

( = =3 2

3 3 3 2

2 2 5

5 5 5

) = = =4 4 4 3 2

2 2 2 2

3 3

) = =3 3

3 2

2 7 2 14

22

(x

xx x(x )

= =

>

2 2 2

0

(

z(z )

=

3

3 2

3

5

0


84/151

.

(3

6

2

(2

32

)

12

6

) x3

5

3

)x(

.

1

( + +2 50 32 2 72 )3 227

( ( )( ) +2 5 10 2

( + 8 128 50

(

3 27

64

( 2 48 3 27

x x22 .x>0 2

3 5

.

a

a

h

.

.a

ABCD Q P N M

ABCD .

MNPQ

aa

h

a

2

a

2

MA B

Q

D P C

N


85/151

:< = > 6

+5 4 +5 4 4 +2 2

3 2

3

11

3

11 +2 23 4 5

: 7

) = 10 ) =2 6 ) =1

3) =8 2

(

=

52

64

2

)( )

=

2

2

123

4 3

)m m

m

m

=

6 2

)

( )

=

3

3

39 27

4

. 8

)5

2 3)

a3 2

2)

2

7

. x ( x )=2 2 9

x0

)

.

)

.

)

.

.

10-9

.

.


86/151

a

b

c

a = b +c

2 2 2

.


87/151

.

.

7 , x , x, a x z 3 23 ,1

5xy , x, z ,

2

7

.

x

1 , 3x, x2 , |x| , x+x , y3 , +x

-3 xy x y27

2 xy

.3x3x

:

( )- x7 x( = )-8 x3(=- x3

( ( x y) ( ) .(x ) . y x y= =2 3 3 2 3 3 6 32 2 8

3 3 27

3( ( x ) ( x ) 3 2 2 31

33

=

( ( a b)(ab)( a c )

=2 2 51 2

2 7

( ) xy()- xy(=

( )xy()3xy3( + xy3)- x3y(=

a

ax3y

. y 3 x a

.

+3 =

y

x

:


88/151

.

x

yy

x

2+4=642a , x , ya x y3 2 43

5x2y2z4

-12xu

3

5

.

-5 x3y2

3x2y3

)

(

: .

x-x+ , x-x ,

ax y axy axy 2 22 3

3 2 , 3x

x

- xy3+ xy-

3. y

. x y .

)

(

.

:x 3

( 3x+ -x + x3= x3+3x-x+

( -3bxy3+axy-bx3y

( x y xy x y + 2 2 3 31

2 3 42

.

x

( -a-3ax+x-]a+ax-)3a-8ax([=


89/151

-( )x+ x()x3 x +( =

( )x+x()x+x+( =

()x-x+( )x+x3-(=

:

x

)x+3(2x2+6x+96xx2x

-20

5

3

2

.

x

.

x+x+9

)x+3(

)x+3(=)x+3()x+3(=

.

.

)x+3(=x+x+9

. x=.3x-3 =x+

.x

.

.

( )a+(=)a+()a+(=a+a+a+=a+8a+


90/151

( )x+(=) + () + ( = + + + =x+x+

( )a+b(=) + () + (= + + + =a+ab+b

)x+( =x+x+

3

:

x x

x x

)x+(

x+

:

( x ) ( x) x+ = + +2 2 25 2 5 2 5 2 2

b

a

)a+b(= a+ab+b .

= )a+b(= s+s3+s

.

( )x-(=)x-()x-( =

=( )3x-x(

=( )a-b(

.

S1 S3

S3 S2

a

a

b

b


91/151

.

a(b+c) = ab+ac

. 1

) (2x+1)2= =) (4a+3b)2

) (x ) =2 21

2

) ( xy x ) =3 21

2 2

) ( )+ 22 3 3 = ( ( ) 25 2 2 =

. 2

( b ) a b+ = + +2 2 2 4

3 4 9

=2a = 4a2=(2a)2

=2(2a)(3b2) = 12ab2

(2a+3b2)2=4a2+12ab2+9b4 :

) (1+b)

2

=1+ +b

2

)

(xy ) = +

21 1

2 4

( ( ) xx

= +2 4

4

1 (x0) (-) (

2=36x2-12xy+

:

(a+b)2= a2+2ab+b2 : b a

(a-b)2= a2-2ab+b2


92/151

)a+b( a+ab+b

:

a+ab+b=)a+b(=)a+b()a+b(

.

( x+x+9=)x+3(=)x+3()x+3(

)x( )3(

=( x-x+=) - (

=( n-n+=) - (

( 8ax+axy +8ay=a )x+xy+9y( =a) + (2=

)

.

.

(=a

ab+ac

:

ab+ac=a)b+c(

)) )..(

:

( 8x+x= x.x +x.3

)

.

.

(=x

= x) x + 3 ( )

(

( a-8a3=

( 7x-x3+x=

= ( xy-xy+xy


93/151

.

( ( m) ( m) ( m) ( m) 2 3 2 31

5 2 22

)( 7a3-b3+c3-)a3-9b3-c3

)( )xm-()xm- ])( x-])y-x(-)y-

.

2

(

(

. 3

( )y-3x(= =( )-3a-a(

) ( x ) 21

83

= =( )/7(+)/7()3/3(+)3/3(

.

( )x+y(-)x-y(= 4 x y

( a (a )aa

+ = + 2 2

2

1 12 )a(

.

( x3+8x+8x ( 3a3b-ab3+a3b3

( a)x+(+b)x+( ( a3-a+a

( xy-xy+

( x+3x3+9x

.

)a+b(=a+ab+b-b b

(a b )

(a ( b))

+ =

2

2

3m2n

3m2n

2m2n

3m2n

2m2n

5m2n

3xy

4xy

5xy


94/151

. 1

=:)a+b+c(2 )a+b+c()a+b+c( =

:))a+b(+c(2= )a+b(2+2)a+b(c +

.

)a+b-c(2=

.

2

)a+b()a-b(=

)a+b()a-b(=a2-b2 b


95/151

.

1( )10-x()x+10( =)10-x()10+x( =100-x2

2( )-y-2z( )-2z+y( = )____- y( )-2z+y( = 4z2-_____

3( )-7y+t()t+7y( =

4( )-4y-2z()2z-4y(=

5( )x-2y+5( )x+2y-5( = [x- )2y-5(] [x+)2y-5(]=

.

4x2-)7-3y(2

.

)2x-7+3y()2x+7-3y(

.

)A2-B2=)A+B()A-B

.

1( x2-9= )x+3() - (

2( y z2 41

44

= ) + () - (

3( )2x+1(2- y2=[)2x+1(-____][) (+y]

4( 1-)3a+z(2

=[1-) (][

+) (]=) ()1+3a+z(

5( )2x+1(2-)3x+4(2=[) (-) (][) (+) (]

=)-x-3() + (

6( x4-y4=)x2+y2() - (

=)x2+y2()x+y() - (


96/151

.

.

1

( )x+2()x+5(=x2+5x+2x+10=x2+7x+10

2+5 2*5

( )x+9()x-4(=x2+9x-4x-36=x2+5x-36

.

)x+a()x+b(=x2+) (x+______

.

1

2

b a

x2+)a+b(x+ab=)x+ ()x+ (

( x

2

+7x+10=)x+ () + (

( x2+7x+12=)x+ () + (

(=( y2+y-6 + () - (

(=( y2-y-6 + () - (

( y2+5y+6=) + () + (

B A

.

.

98*102=)100-2(*)100+2( =______=

497 * 503 =______*______=

)1001(2= )1000+1(2=______


97/151

x2+10x-24 3

.

:)x+6()x-4(

:)x+12()x-2(

:)x+6()x+4(

:)x-12()x+2(

. 1

( ( x)( x) +1 1

4 4) )3x+y-z()3x+y+z(

( )5x+4( )5x+3(

( )x-1()x+1()x2+1(

)

(z )(z ) +3 3 )( )x-2( )x+2( )x+3

. 2

( )xy-z()xy+z(=-z2

( )x+a()x-b(=x2+.-

) ( )( ) y+ = 21

5 54

-( )x2+()x2-5(= x4+2x2

.

3

( a2-8a+15

( x2-13x+36

( x2+x+1__4

)

x2-12x+36

)

x2+10x+24 ( )x+y(2-9

( x2-2x-8 )

bx2-5bx-50b

( 4ax2-a )

x4-5x2+4

b

a

a=b

4

5

.

x2

ax

bx

x + a

x + b

ab

x

x

a

b


98/151

.

b

a

.b a

< >

.b a

a b a b b a

p

a=b+p

p

a < b b a

.a=b+p p

.

( x=y+4x b( b a x

1>2>5 :

:

a b

a x b< 3

7 9 21 27

( ) > 1

.p q2)p-1( =2q-3 ( .a ba-b=1 )

.a b

a b

= 32

)

.u v

u-v=-2

)

:c, b, a 2

)ac

b2>0 )

a

bc


102/151

.65 85 7

4500.

(. 232(

.

..

.

)(

.


103/151

.

.

)

(


104/151

.

.

: n2n

, , , , , n

, , n

1 2 3 4

2 4 6 8 2

..

0 1 1/5 2 3 3/5 4 5x )(

0 2y )

(

x

x

y

y

x

:


105/151

y

x

y

x

1

)x(

)y(

1

2

3

4

100

x

y=______

xy y x

.

0 0/5 1 1/5 2 2/5 3x

)(

0 0/25 1

y

(

)

0

0

:

y=-x+

x

y

=

=

1

x

y

=

=

2

x

y

=

= 3

x

y

=

= 4

x

y

=

= 0

.

y= x=


106/151

. 2

.

0 1 -1 2 -2)

( x

2)

( y

0

2

x

y

.

.

x-y= y=x-

x

y

=

=

x

y

=

=

x

y

=

=

x

y

=

=

x

y

=

=

x

y

=

=

x

y

=

=

x

y

=

=

x

y

=

=

x

y

=

=

.

y=ax+b

.

y

x

.

b=

a =

y=x+

.


107/151

.

y=-x+ y x=3

2

)

0

0

(

y=x

y=x

a

y=ax

. y=x y=-x

.y=ax

.

0

0

1

2 (

0

0

1

1 )

0

0

3

1

y=x

.


108/151

. y x= +1

42

.

2

1)

.)

.

-

)

60

20

6

3

6

2

)

(

.

y=x-

y x

y

=

=

2 1

2 2 1

. y x= +1

22

-

.y=x+


109/151

.y

x .

.

y=/8x+

.

(

1

3

0

0

)

1

1

2

3)

0

1

1

4

.y=-x+

. y x= +3

45

6

.

y x= +1

22

7

.-1

2

3

.-2

.


110/151

1

.

=( y1

2x ) y=x ) y=3x ) y=-x ( y=-2x

.

x

x

2

.

y=2x-1 , y=2x , y=2x+3

2 x

.

x

a . a y= ax+b

b .

.

.-3 2 y = -3x+2

:

y=x


111/151

. 1

y=2x-4 y x= 2

3y = -3x+1

: 2

.-1 -2 (

.31

2(

.

0

4

y=2x +1

)

.

1

22 3

y ax b y x b b b y= + = + = + = =

2 2 2 1

2 2 1

1

.

2

B A

.

y

x

A

B

O

y

Hx

AH

OH

BH

OH


112/151

y = x - d3

.

B

A

.

d EBEA

= = =2 21

D

C

.

.

.

)1( .

)2( .

)3(

d1

A

B

C

D

F

E

d2

A

B

C

D

F

E

0 4d

.

.d

d

EB

EA= =

0

1

d

.d.1

. y=x-


113/151

2

1

2

0

.

.

:

2

:

.x=

.ax+by = c

)

a= , b= , c=

a=- , b= , c=

.c b a (

-x+y= a= b= c=

y=x+ a= b= c=

.

c

b

a

x=

)

ax by c x+ = =

2

.

A =

B

=

C =

D =

A B C D


114/151

.x = - y=

.

y-x = 6 x-y = 8 x-y =

.

x =

y = -

.

2

1x

.

c

b a

ax by c+ =

.

y x y x y x = = + = + 4 8

2 4 8 2 4 8

2 2

y=x+

x-y=6

x+y-9=


115/151

.

1

1y-x= 6

.7

=y y

x x

2 1

2 1

.

4

1

3

2

8

x2

y2

x1

y1

xA

y

1

1

xB

y

2

2

a

b b

.

.y=ax+b

.


116/151

2000

3000

1

y x .

.x y

3000

x ))y .

.

.

1/5

y=2x-1 2

.

y = -x +2

.

.

:

y

x


117/151

x-6y=

.

x-y=

.

)

)________________________

)

2x - y =

x - 6y = 8

.

x y

x y

=

+ =

1

3

x y

x y

=

+ =

2 3

2 3


118/151

.

(

x yx y

= + =

3

4 2 6(

x yx y

= + =

3 5 1

2 3 7(

x yx y

+ = + =

3 2 50

2 2 35

1kg1kg 1kg 1kg

1kg1kg

1kg 1kg

2kg

2kg

:

(

=

+ ==

2 2

5 5

7 7

(

x x

x xx x

=

+ ==

2 2

3 3

(

_______________

.

.

(

{x yx yx

=+ =+

=

1

3

2 4

)

{ {x y x yx y x yx

= =

+ = + =+

=

2 2 3 4 2 6

2 4 2 4

5 10

x= x+y= x= x+y=

y y+ = =2 3 1 +y =

y y= =2 2 1

:

2

1

:

2

1

y x

.

=

=

=


119/151

.

x y

y x

=

=

2 3 5

1 2

3 3

y : (

.)

x- ) (=

.

.

..

:1

:x

:x-

=)x+x-(=

:2

:x

:y

{ {y x x y(x y) x yy y

= =

+ = + =

=

2 3 2 3

2 24 2 2 24

2 3 24

:3

:x :y

y x

(x y) (x x )

=

+ = + =

2 3

2 24 2 2 3 24


120/151

.

(

x y

x y

=

=

3 7

2 7 15(

x y

x y

=

+ =

3 6

12 8

3

.

(

(x y) y

x ( x y)

+ =

=

2 3 4

3 2 2 7(

x y

x y

=

+ =

1 1 1

2 3 6

4

.y x

22x-y-=x+y-

x+y= x-y=

.

23

ay = ax +

a=-

a=

a=

.

.

6

.

( )

.

6

x y

x y

=

=

2 3 7

4 6 5

.

.

.

6

7

7

.


121/151

.

.

.

L

L+ +

150

2150


122/151

2

3.4

.

x+4 x

: x

x+4

x

x=

+

2

4 3

3x = x+8 x=8 =

. x

x+ 4

.

v

v v

v v+1 2

1 2

2v

:

.

v

k

2

2

k v

a b+

2

b a

:

x

x x

+3 22 5

5 2 1

x

x

+

5

1

a

4

2

5

x 3

4

x

y

x x

xy

+2 3 1

9

x

1

x +

10

2

x

x

+2

3 7

xy

(x y)

2

2

x3

1

a

b

x3+x-

( .(

xy

x

x y+

|x-y|

x

1

2

:


123/151

x

7

1

x+6

3

ah

2

x+3

5

x2

25

| x | | y |

x

+

x y

x

+2

1

x

+

5

3 1

x31

mn n

n

+

2

5

4 a

x

+

3

2

:x

x

+

5

3

-5-101

27-2x

x

x

+

5

3

x

x

+

5

3x=3

.8

0

.

x x

(x )(x )

+

+

27 1

1 2:

:

:

)x-()x+(=

:


124/151

(x ) x

(x ) x

= = + = =

1 0 1

2 0 2

.

x = -

x =

.)(

) y

y y=

3

5 2

18 3

60 10

)(x ) (x )x x

x x

+ ++ + =+ +

2

2

3 36 9

4 3 (x ) (x )+ +1 3

x

x+=+

3

1

)x +8 5

2)

x

x

+7)

b

b

+

2 1

2 1

)x

x +23

4

(x

x 2 1)

a

a a

+

+25

5 6

3

4.

3

418

246

89

12

36

48

:

= =

36 3 12 3

48 4 12 4

ac a

bc b=

(b , c ) 0 0

:AC

BC

AC A

BC B= )B C C B )A


125/151

:

)m

m

216

4)

m

m

+

+

6 18

7 21

)a a

a a

+

2

2

5 14

2

)x y

y x

4 4

a ax

a

+

)a ax

a xa

+= + )

a ax a( x)x

a a

+ += = +

11

.

)x

ab25

3

)y

y( y )

2

2 6)

P

P P 22

12

(

x

x

+2 5

(

x

x

+

21

5

(

a

a

+

+

3

2 1

:

)x

x x

+23

5 6

)x x

x

+

+

24 8

12 24)

x

x x

2

2

24

12 6

( y y yy y

+

3 2

2

2 3

( tt

+

4

2

1

1

(x y

x y

2 216 9

8 6

)y (y )(y )

y (y )

+ = =

+ +

29 3 3

3 9 3 3

(

ab b

a b =

7 4

2 3

8 2

20

(

b b

b (b )

= =

5 5

15 5


126/151

)a b

ab

4 2

8

6

4

)a

a a

+ 2

2 8

2 8

. -1 1 3

(y

y

+

2 3

2 3

(y

y

2 3

3 2

)y

y

+

+

2 3

3 2)

y

y

+

2 3

2 3

.

4

5(

a

a

+

2

54(

a

a

+

+

2

53(

a

a

2

5

2(

a

a

+

2

5

1(

a

a

+

2

5

10(

a

a

+

+

2

59(

a

a

2

5

8(7(a

a

2

5

6(

a

a

2

5

5

)z

z z z

=

+3

1)

x

x x x=

2

3

3 6

( ( )y +

=3 2 1

5 5

(( )(x )

x(x )(x )

= +

51

2 5

. z(x y)

t

+ 6

)z

(x y)t

+ )zx y

t

+) z(x y)

t +

1

(x y

zt

+

(zx zy

t t+

(zx

yt

+


127/151

.

:

a c ac

b d bd =

a c a d ad

b d b c bc = =

)b , d ( )b , c , d(

.

:

.

.

) xy z yz

xx z y =

1 23 3

2 2 2

1 3

5 16 2

38 15

)x x x

x x x

+ + =

2

2

3 3

2 15

x

x (x )

+

2

3 (x )=

5

)x x x x ( )( )

( )( ) (x )(x ) xx x x x

= =

+ + + + + +

2

2 2

6 3 18 6 1

3 4 412 36 7 12

)x x x y

xy xy xy

= =2 2 3

3

4 8 4

3 3 8 6

(a a a a a a a

aa a a a a a

+ + =

+ +

2 2 2

2 2 2

4 5 3 2 4 5 4

44 4 3 2

(a )(a ) a a

a( ) ( )( )

+ = =

1 5 4 5


128/151

.

(

a a a

a a

+

+

2

2

6 3

3 4

( a b ab aba (a b)

+

+

2 2

2

3

)x x x

x x

+ + +

+ +

23 2 1

2 5 )

x x

xy y

4

2 5

4 8

3 9

:

a c a c

b b b++ =

a c ad bc

b d bd++ =

) b ( )b , d(

.

.

.

)x x x x

x x x

+ + + + = =

+ + +

3 7 2 3 3 7 2 3

2 2 2

+ =3 4 7

5 5 5

)x x x ( x ) x

x x x x

+ + + = = =

+ + + +

3 7 2 3 3 7 2 3 10

2 2 2 2

=

(a a a (a )

aa a

+ + =

+

2 2 2

2 2

20 2 20 2

24 4

+ =7 9

10 5 10

a a (a a ) = = =

2 22 4 16 2 2 8

(a )( )

(a )(a )

= =

+

2 4

2 2

)a a (a )( ) ( a )

a a(a ) a(a )

+ + + + = =

+ +

1 3 2 1 3 2

2 2

=

= =


129/151

.

)x y

x y y x+

2 2

)x x

+

6 4

)x x

xx

+

+

2

2

2 16 4

24

)x

x x x x+

+ +2 27

2 4 3

(

x x(x y) y(x )y(x y) x(x y) xy(x y)

+ + ++ = = 2 2

2 1 2 3 1

3 3

.

)

(

)x( .

(

x ( )

x x ( )( )x xx x( )( ) xx xx ( )

x xx x

= = = = + + +

22

2 2

22

2 2

1 6 1 61 1

64 3 4 3 14 3

1 1

)

x xx x x x x xx x x

x x x x xx x x

+ = = =

+ +

2

2 2 22 2

2 2 2 2

2 2

1 6 61

6 4 3 6

4 3 4 31

x

2

x x +2 4 3

= = =

(


130/151

)a( )( )

(a ) aa a a a

a (a )a( )( )

a a a a

+ + +

= = =

+

+ +

2 3 2 3

2 1 31 1

2 3 2 3 2 3 1

1 1

(

) a a

a a

+ =

+

2 3

1

2 3

1

a a-

) ( .

)

nn

n m

m

n m

=

+

2

2

2 21

)

y x

x y x y

x y

x y x y

+ =

++

.

5

26

.

.

= x+5

=( )2

x + 3

x + 5


131/151

=) () (

= =5

26

(x ) x

( )( x ) x

+ +

= = + +

25 5 5 5

4 2 3 26 4 6 13

7x=35x=5

=

=

)x


132/151

.

)x y(x y)

x y x y

+

2

2 2 4 4

)x y

ax bx by ay

+ ++

7 9

)a b a b

a b a b

2 2 3 3

2 2)

x xx

x

+

+

24 2

22

. 3

a

a

+2

2

5

a +2 3

3

a b

a

+2 2

2

a a

a

2

5

. x

y 4

)x

y

+

+

3

3)

x

y

3

3 )

x

y

3

3

)x

y

3

3 (

a x

a y

3

3

x

x +

+

3

5

)x

x

+

+

3

5)

x

x

+

3

5)

x

x

+

3

5)

x

x

+

3

5

6

)a a

a(a )(a )a

+ += = + 2

5 55

5 525

)a a

(a )(a ) aa

+ += =+ 2

5 5 1

5 5 525

.A B

C

2 2

2 C=ab B=a+bA=a-b


133/151

. 8

( ..(

(a b a b

b a ab

=

(x

xx =

13

7

20

)a b a b

=7 7

5 5 5

)a b

b a

=

1

(a b a b

=

+

1 1

) a b a ba b

= +

2 2

(

ca cb a b

c cd d

+ +

=+

(

a

cba b

c

=

. 9

.)(

.

)

a a

aa

aa

+

2

2 1

1

) x y x y

x y

(x y)

+

2 2

2

1 2

9

)x

xx x x+

++ + 2 22 1 2

12 1 1

:

.

a

a

+

2

7

)

.a

a

+

2

7)

.x

. x-9

x2 - x - 12_________

x-4

A = x2 -9


134/151

) 1

)

.

x y x

yx y=

5 3

2 2

14 7

2

a xz a z

x z x

=

2 4 2 3

6 5

18 2

27 3

2

:

d

c

b

aa b c a b c

d d d d

+ += + +

:6 x3-8x+

x x x xx x

+= + = +

3 3

312 18 2 12 18 2 12 3

6 6 6 6 3

:

.

)a a a a a a

aaa

+ = + = +

4 3 4 3

2

2

2 5 8 2 5 8 1 2

24

)x yz xy x y z

x y z

+ = +3 2 2 2

2 2

14 6 3

2

x z

y= +

7 3 3

2

)y y y

( y y y) ( y )y

+ + =

3 2

3 2 2

2

8 4 128 4 12 4

4

y= + = + 2 1


135/151

.

)a b c

ab

7 4

6

21

28

)x y z xyz

x z

+6

2

24 2 3

3

-

26 8

24 3

2

:

. 3 8 6

:

+ =


136/151

x22 x x

x

+ 2

7 15 5

2 x x

x

10 2

3 15

- +

x- 3x-

. x- 3x-

x

x=

33 :

.x x x

x x x

x

+

2

2

2 7 15 5

2 10 2 3

3 15

x 3 15

0

.

.

.

1

)

x x x x

x x

x x

x

+ +

+ +

+

+

3 2 2

3

2

2

4 3 7 2

4 8 3

3 9 7

3

1

)

x x x

x x

xx

+ + +

+ +

2

2

5 24 8

8

3 24

3 24

0


137/151

..

x x x x + +4 2 210 3 2 19 3 2

x x x x

x x x

x x

x

x

+

+ + +

+

+ +

4 2 2

4 2 2

2

2

10 3 2 19 2 3

10 15 5

12 2 19

12 18

2

.

) x x x x + 3 26 19 16 4 2 ) x x x + +3 12 8 6

.

)x y z

xz

2 3 7

5

2

18

)a y a y xy

y

+

3 4 2

2

2 15

5

)( )x3- )x-( )( )3y-y-4()3y-4

( x x x x x x+ + + +5 4 3 22 5 2 2 2 3 3

x x x x + + + 4 6 2 33 4 5 1 .

.x3+x+8x 3

. x+4 x

.a 4x+3 x3+3x-x+a 4

.

2x

2

-9x+9

5x+3 x-3 x-3 x+3

:

)x+ ()x-3(+x-

=__________________

=x4-3x+x-9

x-

.x-3


138/151

..

.

.

.


139/151

:.

.

: 1

.

.

.

_________________

______

.______._________________

2

.

.

.

.

.

RO

:


140/151

.

R .

.R

:

: :

.

.

.

)

)

.

______

)

)

.R

. V R= 34

3R

.10 1

.

)

.

)

.

)10. 2


141/151

1

.

.

!

))

.

R 2

(. (

____________:

)____________:

R

)

S=4R2:R

R

R

)) 1

.

10

.10 2

.


142/151

.12800 1

.

)

.)

.)( )

.

1/648/000 )

.

2

.

60

.

100

.

3

24

m

60cm

24cm

24cm

x


143/151

.

.

.

.

.

.

.

: :

:: :

.

ABCDE

OH

.

OABCDE.

:

1

.

.)(

: :

:

:

:

)

2

B

A

CD

E

H

O

H

O

A

B

C

D


144/151

ABCD

)A.

B

C

D

( N M

NABCD

MABCD

)( )2(

)3( )4(

.)2( ))1

O

H

< OH

V

V

.)4( ))3

O

H

=OH

SABC < SA

B

C

V V

(

)

)

.

O

A B

C

H

O

A B

C

H

O

A B

C

H

O

A B

C

H

MN

A

C

B

D

A B

D C


145/151

E

C

.

D E F

ABC

. D B D

.

ABC OABC

OH ACB

=

90 BC =10cm AC =6cm

. 5cm

.

A B

O

C

H

BDE ABD ABED 1

CBAD

CBED

2

ABC

DEF

3

DABC

CDEF

4

4 2 5

h : S

V Sh=1

3

C

D

A

E

B

C

E

B

D

C

B

D

A

AB

C

E

F

D

C

E

F

D


146/151

ABC

AC CBS cm

= = = 2

2

V Sh cm= = = 31 1

3 3

.BC M 10cm

OBC

OM

)

OBM (

12cm

)

.OM OBM

MH OMH (

OH

OMH

)

.

.OABCD (

.

.

.

O

A

C

B

MH

D

___ ___

______

___ ___

___


147/151

:

V Sh R h= = 21 1

3 3

h

R

10cm

.12cm

.

cm12

cm10

12cm

4cm

6cm

5

6

1

.10

4cm 2

.

8cm

12cm

cm

3

6cm


148/151

1

.

:

. 2

5

8

4

3

4 3

10

AB

AB

.

)

(

)

(

10

4

4

5

A

O

B


149/151

1

.

a

2

.

1

:

:

.

2

.

: :

3

.

a

3

10

3

10

20

10


150/151

. 5 cm

5

1

.

2

( ) )

. 3

4r o

r

r


151/151

a

a

a a

a

.

.

.

1

a

a

a

a

a

iranian 9th grade coursebook : math

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